Peter Chesson - Univerzita Karlovaweb.natur.cuni.cz/~botanika/herben/succisa/ChessonARES.pdf ·...

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September 25, 2000 10:21 Annual Reviews AR113-15

Annu. Rev. Ecol. Syst. 2000. 31:343–66Copyright c© 2000 by Annual Reviews. All rights reserved

MECHANISMS OF MAINTENANCE

OF SPECIES DIVERSITY

Peter ChessonSection of Evolution and Ecology University of California, Davis, California, 95616;e-mail: [email protected]

Key Words coexistence, competition, predation, niche, spatial andtemporal variation.

■ Abstract The focus of most ideas on diversity maintenance is species coexis-tence, which may be stable or unstable. Stable coexistence can be quantified by thelong-term rates at which community members recover from low density. Quantifica-tion shows that coexistence mechanisms function in two major ways: They may be(a) equalizingbecause they tend to minimize average fitness differences betweenspecies, or (b)stabilizingbecause they tend to increase negative intraspecific inter-actions relative to negative interspecific interactions. Stabilizing mechanisms are es-sential for species coexistence and include traditional mechanisms such as resourcepartitioning and frequency-dependent predation, as well as mechanisms that dependon fluctuations in population densities and environmental factors in space and time.Equalizing mechanisms contribute to stable coexistence because they reduce largeaverage fitness inequalities which might negate the effects of stabilizing mechanisms.Models of unstable coexitence, in which species diversity slowly decays over time, havefocused almost exclusively on equalizing mechanisms. These models would be morerobust if they also included stabilizing mechanisms, which arise in many and variedways but need not be adequate for full stability of a system. Models of unstable coex-istence invite a broader view of diversity maintenance incorporating species turnover.

INTRODUCTION

The literature is replete with models and ideas about the maintenance of speciesdiversity. This review is about making sense of them. There are many commonali-ties in these models and ideas. The ones that could work, that is, the ones that standup to rigorous logical examination, reveal important principles. The bewilderingarray of ideas can be tamed. Being tamed, they are better placed to be used.

The most common meaning of diversity maintenance is coexistence in the samespatial region of species having similar ecology. These species are termed here “thecommunity,” but are in the same trophic level and may be described as belong-ing to the same guild, a term that commonly means species having overlapping

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resource requirements (124). Another meaning of diversity maintenance refers notto coexistence of fixed sets of species but to the maintenance of species richnessand evenness over long timescales, necessitating consideration of speciation andextinction rates, and infrequent colonizations (35, 74). The primary concern of thisreview is with diversity maintenance as species coexistence.

Many models of species coexistence are thought of as models of coexistence insome defined local area. However, to make any sense, the area addressed must belarge enough that population dynamics within the area are not too greatly affectedby migration across its boundary (103). At some spatial scale, this condition willbe achieved, but it may be much larger than is considered in most models and fieldstudies. There is a temptation to consider diversity maintenance on small areasand to treat immigration into the area as part of the explanation for coexistence(81), but that procedure becomes circular if immigration rates are fixed and arenot themselves explained. Species that continue to migrate into an area are foundthere even if the habitat is a sink. Continued migration of a suite of species into alocal area depends on diversity maintenance in the areas that are the source of theimmigrants. Thus, nothing is learned about diversity maintenance beginning withthe assumption that migration rates into local areas are constant.

STABLE COEXISTENCE

Species coexistence may be considered as stable or unstable. Stable coexistencemeans that the densities of the species in the system do not show long-term trends.If densities get low, they tend to recover. Unstable coexistence means that thereis no tendency for recovery and species are not maintained in the system on longtimescales. In early approaches to species coexistence, stable coexistence meantstability at an equilibrium point. These days, it is commonly operationalized usingthe invasibility criterion or related ideas (7, 13, 36, 51, 83, 94, 133). The invasibilitycriterion for species coexistence requires each species to be able to increase fromlow density in the presence of the rest of the community. A species at low densityis termed the invader, with the rest of the community termed residents. In calcu-lations, the residents are assumed unaffected by the invader because the invader’sdensity is low. The important quantity is the long-term per capita growth rate of theinvader,r i , which is referred to here as the “long-term low-density growth rate.”If this quantity is positive, the invader increases from low density. This criterionhas been justified for a variety of deterministic and stochastic models (36, 51, 94),and has been shown in some cases to be equivalent to more ideal definitions ofcoexistence such as stochastic boundedness (36). Most important, the long-termlow-density growth rate can be used to quantify species coexistence (27).

For a species to have a positive,r i , it must be distinguished from other speciesin ecologically significant ways (34). Indeed, the question of stable species coex-istence is largely the question of the right sorts of ecological distinctions for thegiven circumstances. The Lotka-Volterra competition model is a useful beginning

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for discussing the basic principles. For this reason, it continues to have an importantplace in text books and also in the primary literature where it has roles approx-imating, interpreting, or describing the outcome of more sophisticated models(37, 104, 127, 130), as submodels (16), with variations on it increasingly fittedto field data (15, 42, 92, 115, 139). Unfortunately, textbooks muddy the water byparameterizing Lotka-Volterra competition in terms of carrying capacities and rel-ative coefficients of competition. In terms of absolute competition coefficients, thetwo-species Lotka-Volterra competition equations can be written as follows (30):

1

Ni· d Ni

dt= ri

(1− αi i Ni − αi j Nj

), i = 1, 2, j 6= i . 1.

The quantitiesαi i andαi j are, respectively, absolute intraspecific and interspecificcompetition coefficients, and the defining feature of Lotka-Volterra competition isthat per capita growth rates are linear decreasing functions of the densities of eachof the species. Parameterized in this way, speciesi can increase from low density inthe presence of its competitor resident in the community ifα j j >αi j ,which may beread biologically as “speciesj cannot competitively exclude speciesi if the effectthat speciesj has on itself is more than the effect that speciesj has on speciesi.”The criteria for species coexistence in a two-species system are thereforeα11>α21

andα22>α12, which can be read very simply as “intraspecific competition mustbe greater than interspecific competition.” This criterion is equivalent to requiringthe relative coefficientsβ of Bolker & Pacala (16), which equalαi j /α j j , to be lessthan 1. Unfortunately, the usual relative competition coefficients of textbooks (1),which equalαi j /αi i , are not instructive because they compare how the growth of aspecies is depressed by the other species with how much it depresses its own growth.

Per capita growth rates are linear functions of density in Lotka-Volterra mod-els, which makes Lotka-Volterra models special and may bias their predictions(1, 30, 68, 121). However, models with nonlinear per capita growth rates can bewritten in the form of Equation 1 by making the competition coefficients func-tions of density [αi j = fi j (Ni , Nj )], and the results above remain true, providedthe competition coefficients are evaluated for the resident at equilibrium and theinvader at zero.

Lotka-Volterra models and their nonlinear extensions may be thought of asmodels of direct competition (56): Individual organisms have immediate directnegative effects on other individuals. However, they may also be derived frommodels with explicit resource dynamics (104, 120). Such multiple interpretationsmean that these models are phenomenological: They are not defined by a mecha-nism of competition. A mechanistic understanding of competition usefully beginswith Tilman’s resource competition theory in which species jointly limited by asingle resource are expected to obey theR* rule (127). For any given species,R* is the resource level at which the species is just able to persist. The winnerin competition is the species with the lowestR* value. A species’R*, however,reflects not the ability of members of the species to extract resources when they

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are in low concentration, but their ability to grow and reproduce rapidly enough,at low resource levels, to compensate for tissue death and mortality, which areaffected by such factors as grazing and predation. A species experiencing lowgrazing and mortality rates will have a lowerR* than other species, all else beingequal (30). In essence, it is the overall fitness of a species that leads to itsR* value.

Tradeoffs play a major role in species coexistence: Advantages that one speciesmay have over others are offset by compensating disadvantages (129, 136). How-ever, examined in relation to theR* rule there is clearly more to stable coexistence.With just a single resource as a limiting factor, tradeoffs may make theR* values ofdifferent species more nearly equal, but that would not lead to stable coexistence.This conclusion is amply illustrated in the case of a special linear form of thisresource limitation model (13, 30, 141) where the long-term low-density growthrate of an invaderi competing with a resident,s, is

r i = bi

(µi

bi− µs

bs

), 2.

with theµs representing mean per capita growth rates of the species in the absenceof resource limitation, and thebs representing the rates at which the per capitagrowth rates decline as resources decline in abundance (30). In this system, theratiosµ/b measure the average fitnesses of the species in this environment, andthey have the appropriate property of predicting the winner in competition (13, 30):The species with the largerµ/b is the winner (has the smallerR*). Tradeoffs maylead to similar values ofµ/b for different species. However, such similarity inaverage fitness does not lead to stable coexistence as Formula 2 necessarily hasopposite sign for any pair of species, meaning that only one of them can increasefrom low density in the presence of the other.

Coexistence with resource partitioning contrasts with this. Using MacArthur’smechanistic derivation of Lotka-Volterra competition (6, 26, 104), the per capitagrowth rate of an invader can be written in the form

r i = 1

Ni· d Ni

dt= bi (ki − ks)+ bi (1− ρ)ks, 3.

where theks correspond to theµ/bs andρ is a measure of resource-use overlapof the two species. [Theks are theh′ −m′ of (30), andbi is bi

√aii of (26, 30).]

The first term on the right in Equation 3 is the average fitness comparison ofEquation 2 and therefore has opposite sign for the two species. Whenever,ρ < 1(resource overlap is less than 100%) the last term in Equation 3 is positive for bothspecies. This last term is therefore a stabilizing term that offsets inequalities infitness expressed by the first term. The two species coexist if the stabilizing termis greater in magnitude than the fitness difference term because then both havepositive growth as invaders. Alternatively, note thatαis/αss = (ks/ki )ρ, whichmeans that to satisfy the coexistence requirement that intraspecific competitionexceed interspecific competition,ρ must be less than 1, and the smallerρ is,

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the easier it is to satisfy the condition, i.e. the larger the difference ink valuescompatible with coexistence (26, 30).

The stabilizing term in Equation 3 arises through a tradeoff in resource use.The assumptions of the model entail that doing well on some resources means do-ing less well on others. Each species has density-dependent feedback loops withits resources that limit it intraspecifically and limit other species interspecifically.However, limited resource overlap and tradeoffs in resource benefits mean thatintraspecific limitation is enhanced relative to interspecific limitation. This con-centration of intraspecific effects relative to interspecific effects is the essence ofstabilization. Tradeoffs associated not with resource use, but for example, withmortality rates (30, 68), may minimize the fitness difference term, making it easierfor domination by the stabilizing term, but such tradeoffs cannot create stabilityalone.

Equation 3 generalizes to multispecies communities involved in diffuse compe-tition, that is, where competition between species involves comparable interactionstrengths for all pairs of species. In a variety of different models of diffuse compe-tition (including stochastic models), the following common approximation to thelong-term low-density growth rate is found:

r i ≈ bi (ki − k)+ bi (1− ρ)Dn− 1

, 4.

wheren is the number of species in the system, theks are again measures of fitness ofindividual species,k is the average fitness of residents (the competitors of speciesi),ρ is again niche overlap, but not necessarily strictly defined in terms of resourceuse (27), andD is a positive constant. Like Equation 3, the first term is an averagefitness comparison and the second term is a stabilizing term. Without this term, thefirst term of necessity leads to loss of all species but the most fit on average, whichin the context of Tilman’sR* rule would be the species with the lowestR* value.However, if the stabilizing term is larger in magnitude than the relative averagefitness term for the worst species, then all species coexist. These two general termsmay involve different mechanisms. Those reducing the magnitude of the fitnessdifference term will be referred to asequalizing mechanisms, while those increas-ing the magnitude of the stabilizing term will be referred to asstabilizing mecha-nisms. In the absence of the stabilizing term, equalizing mechanisms can, at best,slow competitive exclusion; but in the presence of stabilizing mechanisms, equal-izing mechanisms may enable coexistence. Stabilizing and equalizing mechanismsare concepts applicable beyond diffuse competition, but their implementation andtheir sufficiency may differ for different competitive arrangements. For example,invasion of species occupying a one-dimensional niche axis means that neighbor-ing species on the niche axis would be most important to the invader (105, 112),whereas the diffuse competition formula (Equation 4) implies that only the num-ber of species and their average fitnesses matter to the invader. In addition, somemechanisms, as we shall see below, have both stabilizing and equalizing properties.

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The key question to be addressed below is how mechanisms with stabilizingproperties arise in various situations. The theoretical literature supports the con-cept that stable coexistence necessarily requires important ecological differencesbetween species that we may think of as distinguishing their niches (34, 95) andthat often involve tradeoffs, as discussed above. For the purpose of this review,niche space is conceived as having four axes: resources, predators (and othernatural enemies), time, and space. In reality, each axis itself is multidimensional,a feature that does not intrude on the discussion here. A species’ niche is not aHutchinsonian hypervolume (95), but instead is defined by theeffectthat a specieshas at each point in niche space, and by theresponsethat a species has to eachpoint. For example, consider the resource axis. A species consumes resources, andtherefore has an effect on resource density (54, 95). Individuals of a species mayalso reproduce, grow, or survive in response to resources (54, 95).

The essential way in which stabilization occurs is most clearly seen with re-source competition. If a species depends most on a particular resource (strongresponse), and also reduces that resource (strong effect), then it has a density-dependent feedback loop with the resource and is limited by it. If a second specieshas a similar relationship with a different resource, then even though the specieseach consume some of the resource on which the other depends most strongly(limited resource overlap), each species depresses its own growth more than itdepresses the growth of other species (60, 127). The result is stable coexistence.This conclusion, however, depends on explicit and implicit assumptions, whichif varied, alter the conclusion. A symmetric situation in which each resource isequally rich and each species is equally productive, would lead to identical averagefitnesses, and therefore Equation 3 would have only the stabilizing term. However,asymmetries, in which one resource is much richer (127), one species producesmore per unit resource, or has a lower mortality rate (30, 68), would mean thatki − ks would not be zero, and if sufficiently large, this fitness difference wouldcounteract the stabilizing effect of low resource overlap, causing competitive ex-clusion. This result would occur because the advantaged species would be at suchhigh density that it would consume too much of the resource on which the otherspecies depends. An equalizing mechanism would then be necessary to reduceki − ks before coexistence were possible.

Another implicit assumption in the argument above is that the resources haveindependent dynamics apart from consumption by common species. However, thatis not true if, for example, the resources are simply two stages of the same foodspecies (17) or light intercepted at different heights in a forest canopy (31, 89),requiring special conditions to allow each species to be partially independentlylimited by the resource supply on which they depend (17, 89). The idea that speciesmust be somewhat independently limited is critical to their depressing their owngrowth rates rate more than they depress the growth rates of other species. And it iscritical also that this phenomenon involves a density-dependent feedback loop fromthe species to itself, either directly through some form of interference or indirectly,as discussed here, through a resource. Predators and other natural enemies mayalso provide density-dependent feedback loops for their prey (8, 70, 77). Space

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and time may modify such feedback loops applicable to the community as a wholein ways that intensify intraspecific density dependence relative to interspecificdensity dependence.

FLUCTUATION-DEPENDENTAND FLUCTUATION-INDEPENDENT MECHANISMS

Stable coexistence mechanisms may be fluctuation dependent or fluctuation inde-pendent (27). Examples of fluctuation-independent mechanisms are resource par-titioning (6, 60, 118, 127) and frequency-dependent predation (53, 77). Commu-nity dynamics in both of these cases are commonly modeled by deterministicequations that have stable equilibrium points and are sometimes termed “equi-librium mechanisms.” However, these mechanisms can function in the presenceof environmental fluctuations. Indeed, incorporating environmental variability inLotka-Volterra models by making the per capita rate of increase fluctuate withtime need not change the conditions for species coexistence in any important way(27, 133, 134). Thus, we can think of the operation of the mechanism as indepen-dent of the fluctuations in the system. The tendency to dismiss such mechanismsbecause populations in nature fluctuate is not supported by the results of models.

Some stable coexistence mechanisms critically involve the fluctuations in thesystem, that is, without the fluctuations, the mechanism does not function. Thus,they are termedfluctuation dependent(27). In the case of temporal fluctuations,these mechanisms can be divided into two broad classes:relative nonlinearity ofcompetitionandthe storage effect.

Relative Nonlinearity of Competitionor Apparent Competition

The per capita growth rate of a population is commonly a nonlinear functionof limiting factors, such as limiting resources [e.g. the light saturation curve ofplant productivity (137)], or predators [e.g. if predators interfere with one another(45)]. Stable coexistence may result from different nonlinear responses to commonfluctuating limiting factors (9, 13, 18, 27, 50, 76, 101, 125). As first thoroughly in-vestigated by Armstrong & McGehee (13), for the case of a single limiting factor,the per capita growth rate of a species takes the form

ri (t) = Ei (t)− φi (F), 5.

whereEi(t) is the maximum per capita growth rate as a function of possibly fluc-tuating environmental conditions (an environmental response),F is the limitingfactor, for example the amount by which resources are depressed below their op-timal values, andϕ i (F ) is the response defining the dependence of the per capitagrowth rate onF (27). The departure of the functionϕ i from a linear function isits nonlinearity, which is measured by a quantityτ (27). For example, a type II

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Figure 1 Per capita rates of growth of two species (a andb) as functions of a common limitingfactor. In the case where the limiting factor is shortage of a resource, larger F means less resource,and curvea with positiveτ would be generated by a type II functional response, while curvebwith zeroτ would occur with a linear response.

functional response to a limiting resource gives a positiveτ , while a linear func-tional response gives a zeroτ (Figure 1). Two species with different values ofτmay coexist stably provided the species with the larger value ofτ (a) has a meanfitness advantage in the absence of fluctuations in the limiting factor and (b) expe-riences lower fluctuations in the limiting factor when it is an invader than when it isa resident. Armstrong & McGehee showed that condition (b) may arise naturallywith resource competition because a large value ofτ may cause resources to havelarge-amplitude cycles over time.

These conditions for stable coexistence can be understood from the followinggeneral approximation to the long-term low-density growth rate of a species,i, inthe presence of its competitor,s:

r i ≈ bi (ki − ks)− bi (τi − τs)V(F−i ), 6.

where the first term is the fitness comparison in the absence of fluctuations in the

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limiting factor,τi − τs measures relative nonlinearity, andV(F−i ) is the varianceof the limiting factor calculated for invaderi and residents (27). In contrast toEquation 3 for the Lotka-Volterra model, the second term of this equation is notpositive for both species but changes sign when resident and invader are exchanged.The species with more positive nonlinearity (larger value ofτ ) is disadvantagedby fluctuations in the limiting factor. However, if it is also the species with higheraverage fitness as measured by the first term, then this mechanism can have anequalizing role decreasing the advantage that a species gains from the first term.The fact thatV(F−i ) may vary depending on which species is the invader meansthat it is possible for this mechanism to have a stabilizing role too. An algebraicrearrangement of Equation (6) clarifies these stabilizing versus equalizing roles.DefiningB to be the average ofV(F−i ) for each species as invader andA to behalf the absolute value of the difference between these values, and assuming thatthe species with the smallerτ experiences largerV(F) as an invader, then

r i ≈ {bi (ki − ks)− bi (τi − τs)B} + bi |τi − τs|A. 7.

The term in braces becomes the equalizing term—it is the relative fitness taking intoaccount fluctuations in the limiting factor, and has opposite sign for the two species.The final term has the same sign for both species and is therefore a stabilizingterm. Thus, relative nonlinearity (nonzeroτi − τs combined with variance inF )can have both stabilizing and equalizing roles. However, the stabilizing role mightbe negative, viz destabilizing if the species with the larger value ofτ also has thelarger value ofV(F−i ), as thenA in Equation 7 must be replaced by−A, whichmoves both species, long-term low density growth rates closer to zero promotingcompetitive exclusion (or apparent competitive exclusion ifF relates to commonpredation on the two species).

Although there is a generalization of Equation 6 to multiple limiting fac-tors (27), relative nonlinearity in the presence of multiple limiting factors hasbeen poorly investigated. Huisman & Weissing (76), however, have demonstratedenormous potential for this mechanism with multiple competitive factors. For thedynamics of phytoplankton in lakes they use a model of competition for essentialresources (98, 127) that has strong nonlinearities in a multidimensional sense. Fluc-tuations in these essential resources driven by their interactions with phytoplanktonspecies appear to stabilize coexistence very strongly. This result contrasts with thefeeble effects of relative nonlinearity on coexistence of phytoplankton previouslydemonstrated for the case of a single limiting factor (57–59).

The Storage Effect

Models imply that temporal environmental fluctuations can have major effectson population and community dynamics. It has been widely noted that negativeenvironmental effects may reduce population densities and therefore reduce the

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352 CHESSON

magnitude of competition (55, 78, 79), but this possibility does not necessarilytranslate into less competitive exclusion (30). Instead, models in which stable co-existence results from environmental fluctuations are models of temporal niches:Species are not distinguished by the resources they use but by when they aremost actively using them (2, 12, 30, 102). For stabilization to result, intraspecificcompetition must be concentrated relative to interspecific competition, and thisrequires three important ingredients, whose collective outcome is referred to asthe storage effect(27). The most obvious of these isdifferential responses to theenvironment. Species from the same community may have different responsesto their common varying environment. Environmental variation may be regu-lar and deterministic, for example seasonal (12, 37, 47, 60, 86, 91, 102, 109, 140)or stochastic, attributable, for example, to weather on a variety of timescales(2, 27, 30, 32, 33, 36, 37, 51, 52, 62, 112, 123).

How can differential responses to the environment concentrate intraspecificdensity dependence relative to interspecific density dependence, and hence act asa stabilizing mechanism, when the physical environment is not altered by popula-tion densities? The answer is that population responses to the physical environmentmodify competition, as measured by the second essential feature of the storageeffect,covariance between environment and competition(30, 33). Just as negativeenvironmental effects may decrease competition, positive environmental effectsmay increase it. Covariance between environment and competition is measured bycalculating the standard statistical covariance between the effects of the environ-ment on the per capita growth rate of a population (the environmental response)and of competition (both intraspecific and interspecific) on the growth rate (thecompetitive response).

With covariance between environment and competition, and differential re-sponses to the environment, intraspecific competition is strongest when a speciesis favored by the environment, and interspecific competition is strongest when thespecies’ competitors are favored. The final ingredient,buffered population growth,limits the impact of competition when a species is not favored by the environmentand is therefore experiencing mostly interspecific competition. Buffered popula-tion growth may result from a variety of life-history traits: seed banks in annualplants (29, 48, 49, 113, 114), resting eggs in freshwater zooplankton (22, 62, 63),and long-lived adults in perennial organisms (38, 86, 91, 96, 97, 112, 119). Organ-isms not having specific life-history stages with a buffering effect may be bufferedin other ways. For example, desert rodents and herbaceous plants may have timesof the year when they are dormant and therefore relatively immune to unfavorableenvironmental and competitive conditions (19, 37). Alternatively, subdivision ofa population into different phenotypes, or local populations in space, with differ-ent exposure to environmental effects and competition, may also have a bufferingrole (33).

By diminishing the effects of interspecific competition when a species is notfavored by the environment, buffered population growth, combined with the othertwo ingredients of the storage effect, leads to concentration of intraspecific effects

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on population growth relative to interspecific effects, which is stabilizing. In avariety of models of diffuse interactions involving the storage effect, the long-term low-density growth rate can be given approximately in the form

r i ≈ bi (ki − k)+ bi (1− ρ)(−γ )σ 2

n− 1, 8.

where the symbols follow those in the general diffuse competition formula (Equa-tion 4), with D = (−γ )σ 2 (27). Note that (−γ ) measures buffered populationgrowth and is positive for buffering. The variance in the environmental responseis σ 2, andρ is the correlation between the environmental responses of differentspecies. This formula has a fitness comparison (here representing an average overall environmental states), which is opposed by a stabilizing term (here due to thestorage effect) promoting stable coexistence.

Mechanisms in Combination Integrated over Temporal Scales

Two general factors, competition and the physical environment, drive the mecha-nisms above. Two-factor analysis of variance shows how the effects of two factorscan be divided into their separate or “main” effects and their interaction (126).Applying this technique to population growth (27) expresses the long-term low-density growth rate as

r i ≈ r ′i −1N +1I , 9.

wherer ′i is the effect of fluctuation-independent mechanisms, plus mean fitnessdifferences (i.e. everything that is independent of fluctuations),1N is relativenonlinearity of competition, and1I is the storage effect. Formulae forr ′i , 1N,and1I can be found in Chesson (27), but examples of them are respectivelyEquation 4 and the second terms of Equations 6 and 8. The storage effect,1I,is the interaction between fluctuating environmental and competitive responses,which might serve as its formal definition. The other two terms arise from the maineffects with adjustments to remove all fluctuation-independent effects from1Nand1I, and place them inr ′i .

Although the derivation of Equation 9 in (27) involves approximations (hence“≈”), these approximations occur in the formulae for the components, not thedivision into three terms. The important assumption limiting the generality of theresults is that there should be fewer limiting factors than community membersso that competition experienced by a low density invader can be expressed as afunction of competition experienced by the resident species. With this one caveat,Equation 9 shows that in models of temporally fluctuating environmental andcompetitive effects, three broad classes of mechanisms (fluctuation-independentmechanisms, relative nonlinearity, and the storage effect) are exhaustive: Anymechanism of stable coexistence must be one of these or a combination of them.

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354 CHESSON

Although the storage effect and relative nonlinearity may appear to be rather in-volved, there are no simpler fluctuation-dependent stable coexistence mechanismswaiting to be discovered. In particular, no credence can be given to the idea thatdisturbance promotes stable coexistence by simply reducing population densi-ties to levels where competition is weak (30, 141). More sophisticated views ofdisturbance (23–25, 39, 40, 64), however, have yielded important hypotheses, asdiscussed below under the spatial dimension.

It is now commonly believed that spatial and temporal scale have major effectson the perception and functioning of diversity maintenance mechanisms (10, 1443, 44, 74, 78, 80–82, 100, 116, 143). An important finding is that fluctuation-dependent mechanisms can give community dynamics indistinguishable fromthose of fluctuation-independent mechanisms when viewed on a longer timescalethan the period of the fluctuations (37, 91), as is illustrated in Figure 2. Simi-lar effects are often apparent in spatial models (16, 46, 64, 84, 122), although notalways explicitly noted by the authors. In this regard, the termr ′i in Equation 9can be regarded as containing not just fluctuation-independent mechanisms, butalso fluctuation-dependent mechanisms on timescales shorter than the fluctuationsconsidered in the1N and1I terms. When this is done, Equation 9 can be viewedas an iteration allowing the effects of fluctuation-dependent mechanisms operat-ing on different timescales to be combined to give their total effect on very longtimescales (37).

THE SPATIAL DIMENSION

Nature is strikingly patchy in space. Naturally, if species live in different habi-tats and have no direct or indirect interactions with each other, they should haveno difficulty coexisting in a region combining these separate habitats. However,species do not have to be strictly segregated in space for regional coexistence(16, 20, 38, 90, 93, 99, 112, 122, 123). Spatial variation is similar to temporal vari-ation in its effects on species coexistence with some important differences (32, 38,123). In particular, there is a spatial analog of Equation 9 expressing growthfrom low density in terms of fluctuation-independent mechanisms, spatial relativenonlinearity, and the spatial storage effect (28a). In addition, there is a fourthterm in the spatial analogue of Equation 9 involving the spatial covariance oflocal population growth with local population density, which behaves like a spa-tial storage effect in some circumstances, but like spatial relative nonlinearity inothers (28a).

Spatial storage effects commonly occur in spatial models when spatial environ-mental variation is included. There are two common ways in which the requirementof differential responses to the environment is met in such models. First, relativefitnesses of different species may vary in space (21, 32, 38, 85, 107, 112, 122, 131).For example, for plant species, the identity of the species with the lowestR*value may vary spatially due to differential dependence ofR* on spatially varying

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physical factors, such as temperature and pH (131). Second, there may be rela-tive variation in dispersal into different habitats, for example, in marine habitatsdue to the complexities of spawning, currents, and developmental interactions(21, 32, 42, 85) and in insects potentially due to spatially varying habitat prefer-ences (34, 84, 88). Such relative variation satisfies the requirement for differentialresponses to the environment.

Buffered population growth automatically occurs when populations are subdi-vided in space over a spatially varying environment (32, 38), but the strength of thebuffering is affected by other aspects of the biology of the organisms, includingtheir life-history attributes (28a). Covariance between environment and competi-tion naturally arises when dispersal varies in space but not necessarily when relativefitnesses vary in space. The presence of covariance is easy to determine in a model,and is a powerful tool for determining if coexistence can be promoted by spatial en-vironmental variation. For example, in models of sessile marine invertebrates withcompetition between adults and new settlers, but not among adults, spatial variationin adult death rates leads to strong covariance between environment and compe-tition if the environmental differences between localities remain constant overtime. Then adult densities build up in low-mortality locations, leading to strongcompetition from adults in such localities. Thus, covariance between environ-ment and competition occurs, and regional coexistence by the spatial storage ef-fect is possible (32, 107). However, if environmental differences between localitiesconstantly change with time [i.e. environmental variation is spatio-temporal ratherthan purely spatial, sensu Chesson (32)], local population buildup occurs at bestweakly before the environment changes. Covariance between environment andcompetition is weak or nonexistent, and so the storage effect is weak or nonexis-tent (32, 107).

There has been little explicit attention to relative nonlinearity of competitionin spatial models, but Durrett & Levin (46) demonstrate coexistence in a spatialcompetition model dependent on spatial variation and different nonlinear competi-tion functions that correspond biologically to different competitive strategies. Thespatial Lotka-Volterra model of plant competition of Bolker & Pacala (16) does nothave relatively nonlinear competition, as the per capita rates are linear. However,a similar nonlinear effect arises from covariance between local population densityand competition. This covariance is due to chance variation in local populationdensities, which covaries with local competition in ways that give advantage, un-der some circumstances, to species with short-distance dispersal strategies. Similarphenomena have been found in other spatial competition models (93).

There are many other spatial models of competitive coexistence, few of whichhave been investigated within the framework presented here or a related frame-work (66, 89, 106, 117). Nevertheless, most do have some of the key elements ofthis framework, including nonlinearities, spatial covariances, and buffered growthrates, which appear actively involved in model behavior. From these other mod-els, two ideas emerge as especially important: Tilman’s resource-ratio hypothesis(127, 128), and colonization-competition tradeoffs.

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The resource ratio hypothesis postulates that the ratio of the rates of supply oftwo limiting resources at a particular locality determines a unique pair of plantspecies from a given regional pool able to coexist at that locality. Given variationin space in the supply rate ratio, different pairs of species would be able to coexistat different localities in the absence of dispersal between localities. However,a full spatial model exploring this idea in the presence of dispersal has neverbeen developed. Moreover, Pacala & Tilman (112) pointed out that under certainconditions, a given resource ratio may determine a particular best competitor thatwould dominate that site. With site-to-site variation in the resource ratio, thisscenario leads to coexistence as a simple example of a spatial storage effect (112).Although there would be only one best species at any given locality, as determinedby the resource ratio there, many species could be present at a locality due todispersal from other localities.

Competition-colonization tradeoffs have been discussed in two distinct sorts ofmodels. In one, a patch in space supports a local community that is destroyed atrandom by disturbance. It is then recolonized from other patches (24, 64). In otherinterpretations, a patch supports a single individual organism (87, 110, 130), whosedeath by some means (not necessarily disturbance) opens that locality for recolo-nization. Coexistence in these models requires the colonizing ability of differentspecies to be ranked inversely to competitive ability, which therefore tends to drivea successional process in each locality. Localities becoming vacant randomly inspace and time ensures a landscape in a mosaic of successional states. When vacantsites are interpreted as resulting from disturbance, this model provides perhaps themost satisfactory expression of the intermediate disturbance hypothesis (30), as di-versity tends to be maximized at intermediate values of disturbance frequency (64).

PREDATORS, HERBIVORES, AND PATHOGENS

Predation is a common hypothesis for high biological diversity. However, the ar-gument that predators maintain diversity by keeping populations below levels atwhich they compete is now known to be highly simplistic (4, 30). Predators mayadd density dependence to their prey populations through functional, numerical,and developmental responses to prey (108). In the absence of frequency-dependentfunctional responses or similar complications (65), a common predator of severalprey is a single limiting factor with analogous effects to limitation by a singleresource (67). One density-dependent limiting factor (the resource) is simply re-placed by another (the predator). Indeed, there is aP* rule exactly analogous totheR* rule that says that the species with the highest tolerance of predation willdrive other species extinct (69). Thus, if predation is so severe that it does eliminatecompetition, then competitive exclusion may be replaced by apparent competitiveexclusion.

There are many ways in which predators may help species coexist, but it is farfrom an easy solution (7, 30, 65, 68, 70, 77, 135). Predators may promote species

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coexistence when each species has its own specialist predators, or more gener-ally, specialist natural enemies that hold down the density of each species inde-pendently. This idea, known as the Janzen-Connell hypothesis, is an importanthypothesis for the coexistence of trees in tropical forests (11) where tree seedsand seedlings may be subject to density- or distance-dependent seed predation(41, 138). By providing or attracting species-specific seed and seedling predators,an individual tree would have a greater negative effect on a conspecific growingnearby than on a heterospecific growing nearby (all else equal), thus providing thecritical requirement of a stabilizing mechanism.

Many systems support generalist predators or herbivores, which prey or graze ona range of species (65, 77, 108). Murdoch has emphasized that predators may havefrequency-dependent functional responses (switching; 108), which are stabilizingmechanisms. More complex intergenerational learning responses of parasitoidsmay have similar effects (65). Alternatively, a predator may not be frequency-dependent but have unequal effects on prey species that are also limited by re-sources (135). For example, two competing prey species may coexist if one ofthem is more strongly limited by the resource and the other is more strongly lim-ited by the predator (61). In all of the above instances, the role of predation isto generate feedback loops in which an individual prey species depresses its ownper capita growth rate more than it depresses the per capita growth rates of otherspecies, thus meeting the requirements of stabilizing mechanisms. However, whenthese density-dependent and frequency-dependent requirements are not met, a gen-eralist predator may instead have an equalizing role by inflicting greater predationon the competitive dominant. If a stablizing mechanism such as resource parti-tioning is present, then although the predator is not the stabilizing agent, stablecoexistence may occur in its presence (30).

Predators are often invoked as biological disturbance agents inflicting mortalitypatchily in space and time. When mortality is not species specific, predators may bethe agent of local extinction in the competition-colonization tradeoff models (23).If mortality is species specific, but density independent, then predators may havea role analogous to a spatio-temporally variable environment, which may lead tocoexistence by the spatial storage effect (111). Thinking of predators as organisms,and the environment as the physical environment, invites generalizations of thestorage effect to stable species coexistence in the presence of apparent competition.However, at present there is no theory of covariance between environment and ap-parent competition. As discussed above, predators can be fluctuating nonlinear lim-iting factors, but the potential of this mechanism has not been explored in any detail.

UNSTABLE COEXISTENCE

Hubbell (73, 74) has championed and steadily refined a model of community dy-namics in which coexistence is not stable: The species compete for space butare ecologically identical and therefore have equal fitnesses under all conditions.

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Thus, in Equation 4, both the average relative fitness term and the stabilizing termare close to zero, and the species in the system undergo a very slow random walk toextinction. To many people, a very slow loss of species is equivalent to indefinitecoexistence, and it is certainly one model of how nature is (78, 79, 81, 123). Severalobjections have been raised against it. First, equal average fitnesses seem highlyunlikely (28), and significant violation of this assumption destroys the conclusionof slow extinction (142) by creating nonzero average fitness difference terms forthe species. Minor realistic variations on the model also lead to stabilizing compo-nents (28). Indeed, such features seem difficult to avoid in most systems (28, 142).

Other approaches to unstable coexistence have not assumed the extreme neu-trality of Hubbell’s model, and have sought means by which fitness differencesmay be minimized (78, 79, 81, 123). But these approaches have not recognized thatstabilizing components are difficult to avoid (28), and may have overestimated theeffectiveness of purported equalizing mechanisms (28). Nevertheless, there isundeniable merit in the question of unstable coexistence because it must be thatin many systems at least some species are only weakly persistent because theirfitness disadvantages are comparable in magnitude to the stabilizing componentof their long-term low-density growth rate. At this point, the study of diversitymaintenance needs to take account of macroevolutionary issues such as speciationand extinction, processes (35, 74), biogeographic processes of migration of speciesbetween communities on large spatial scales, and climate change on large temporalscales (37). Hubbell (74) argued that on such large scales, speciation, extinction,and migration processes are dominant, rendering the admitted oversimplificationsof his neutral model unimportant. Independent data on the rates of these criticalprocesses are needed to test this perspective.

NONEQUILIBRIUM COEXISTENCE

Stable and unstable coexistence is one view of the distinction between equilibriumand nonequilibrium coexistence (75). Another view is fluctuation-dependent ver-sus fluctuation-independent coexistence (35). But equilibria are everywhere. Forexample, Hubbell’s (74) model of unstable coexistence nevertheless has an equi-librium for species diversity on the large spatial scale. Thus, it seems best to ask,Is species coexistence stabilized or not? And if it is, Is that stabilization depen-dent on fluctuations or is it independent of fluctuations? The term nonequilibriumcoexistence is better avoided.

LIMITING SIMILARITY

Unstable coexistence requires species to have very similar average fitnesses forlong-term coexistence. Stable coexistence benefits from similar average fitnesses,but requires niche differences between species that intensify negative intraspecific

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effects of density on population growth relative to negative interspecific densityeffects. This requirement of niche differences has often been referred to as limitingsimilarity (3, 112). It is clear that not any kind of niche differences between specieswill do. For example, the discussion of the storage effect above showed that species-specific responses to the environment, which are one sort of niche difference, leadto stability only if they are linked appropriately to density-dependent feedbackloops. This condition applies generally, as has been emphasized here, but has notbeen studied in as much detail for most other mechanisms. The right sorts ofniche differences might be called stabilizing niche differences. A precise limit tosimilarity implies a particular minimum value for stabilizing niche differences. Itis clear there can be no such value (3, 5). For example, in the various equationsabove, the magnitude of the stabilizing term is naturally an increasing functionof stabilizing niche differences (although this feature cannot be expected underall circumstances 6). The smaller the average fitness differences, the smaller thestabilizing niche differences can be. One may be tempted to suggest that averagefitness differences can be zero, but there are plenty of reasons to expect them todiffer from zero in most situations in nature, and as yet we have no theory thatpredicts average fitness differences. Thus, we cannot predict a particular limit tosimilarity, although the concept that niche dissimilarities of the right sort promotecoexistence is generally supported (3).

Models of species on one-dimensional niche axes predict particular limits tosimilarity when one asks whether a species can invade between two particular res-ident’s with a given niche spacing and given average fitness differences (105, 112).Such an invader is automatically at a disadvantage to the residents, however, be-cause it has more competitors close to it than the residents do. Thus, this sort ofanalysis does not seem truly to answer the question of how close the niches ofcoexisting species can be, but it does emphasize that variation in the spacing of theniches of various species in a community, not just the average spacing, is an im-portant factor in species coexistence (5). This feature is not captured by the simpleequations for diffuse competition given here where there is no such variation inspacing.

Recently, particular limits to similarity were also claimed for coexistence bycolonization-competition tradeoffs. However, the limit in question is the differencein colonizing ability of successively ranked competitors (87, 130). This differencehas an equalizing effect because it compensates for a species’ inferior competitiverank. It is not a stabilizing niche difference, but instead reduces average fitnessinequalities that stabilizing niche differences must overcome. There should beno paradox in the idea that dissimilarities in niche differences of the stabilizingsort, which may be characterized as keeping species out of each other’s way,promote coexistence, while differences in average fitness, which determine howmuch “better” one species is than another overall, favor competitive exclusion (14).

An objection that can be raised about all of the above analyses is that they takeno consideration of the increasing sparseness of populations, and the decreasingabsolute numbers of individuals in those populations, as more species are packed

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into a community. Allee effects in sparse (low density) populations (71, 72) andstochastic extinction in small populations (132) both potentially limit how similarthe niches of coexisting species can be when similar niches mean sparser or smallerpopulations. These possibilities deserve further study as they have the uniqueproperty that they would still work when species are equal in average fitness, thatis, they potentially lead to the requirement that stabilizing mechanisms of a certainminimum strength (depending on population sizes supportable in the system) mustexist for coexistence regardless of the strength of equalizing mechanisms.

ACKNOWLEDGMENTS

I am grateful for many very helpful comments on the manuscript from P Abramsand more than a dozen colleagues and students.

Visit the Annual Reviews home page at www.AnnualReviews.org

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Figure 2 Community dynamics on a long timescale for three species coexisting by thestorage effect (fluctuating lines) in the lottery model for intense competition for space(no space is left vacant, 37) compared with an approximating Lotka-Volterra model (37)modified for intense space competition (dashed lines) and the lottery model with no en-vironmental fluctuations (solid smooth curves). The different panels are for mean adultlongevities of 10, 20 and 100 respectively from top to bottom. Comparing the lottery modelwith and without fluctuations demonstrates the necessity of environmental fluctuations forspecies coexistence. The approximating Lotka-Volterra model from (37) shows that thismechanism can be mimicked by fluctuation-independent mechanisms with increasing pre-cision as the longevity of the organism is increased. Lotka-Volterra models with noiseadded (133) give dynamics indistinguishable from the lottery model. The intraspecific andinterspecific interaction coefficients in the Lotka-Volterra model are equal respectively tothe variances and covariances of the environmental responses in the lottery model (37).

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